In a known machine of this kind (DE 27 51 408 A1) the reference axis is defined by a rocker bearing which is disposed on the machine bed and supports a radially projecting beam. The beam is formed with a reciprocating slide guide means along which a reciprocating slide is displaceable radially with respect to the reference axis. A pivot bearing means on the reciprocating slide defines a pivot axis which extends parallel to the reciprocating slide guide means and intersects the reference axis at a right angle. A shaft, with a spindle head secured to it, is supported in the pivot bearing means. Inside the spindle head, a spindle sleeve is guided so as to be displaced by means of a feed drive along a grinding spindle axis which intersects the pivot axis at a right angle. The spindle sleeve, which thus corresponds to a feed slide, includes a grinding spindle adapted to be driven in rotation and carrying a plate-shaped grinding disc for sharpening the toothing of a circular saw blade. When straight surfaces at the front of teeth (cutting faces) or straight surfaces at the back of teeth (clearance faces) are to be sharpened the spindle sleeve always is adjusted by the feed drive in such a way that the active front surface of the grinding disc will lie in a plane which contains the reference axis. For the grinding of oblique surfaces, the spindle head is swung about the pivot axis in one or the other direction, depending on the direction of the inclination. To accomplish that, a cross bar is fastened to that end of the shaft carrying the spindle head which is remote from the spindle head, and two rolls are supported so as to be freely rotatable on this cross bar at equal distances from the pivot axis and diametrically opposed with respect to the latter. These two rolls are arranged between a piston and cylinder unit each and an adjustable stop each in such a way that the cross bar and, together with it, the shaft carrying both the cross bar and the spindle head are swingable back and forth within a pivot range which is limited by the stops. The inert masses thus to be moved are considerable indeed because the spindle sleeve and its feed drive take part in the pivoting motion. Consequently the pivoting frequency to be achieved with back and forth swinging movements is limited.